Suppose $M$ is a Riemannian manifold and $\Phi : M \to M$ is a diffeomorphism. Suppose further that $X$ and $Y$ are smooth vector fields on $M$, and $\nabla$ is the Levi-Civita connection on $M$. Given $p \in M$, consider the covariant derivatives $$ \big(\nabla_X\left(\Phi_* Y\right)\big)_{\Phi(p)} = \nabla_{X_{\Phi(p)}}\left(d\Phi_p Y_p\right) \quad \textrm{and} \quad \big(\Phi_*(\nabla_X Y)\big)_{\Phi(p)} = d\Phi_p\left(\nabla_X Y\right)_p. $$ My question is basically twofold:
- Is there a connection between these two covariant derivatives?
- Is there a canonical definition of the covariant derivative of a diffeomorphism $\Phi : M \to M$, or the covariant derivative of its differential $d\Phi : TM \to TM$?
If $M = \mathbb R^n$, and we write $Y = Y^i \frac{\partial}{\partial x^i}$ and $X = X^i \frac{\partial}{\partial x^i}$, where $X^i, Y^i \in C^\infty(\mathbb R^n)$ for all $i$ (using the Einstein summation convention), then $$\nabla_X Y = (XY^i) \frac{\partial}{\partial x^i} = \left(X^j \frac{\partial Y^i}{\partial x^j}\right) \frac{\partial}{\partial x^i}.$$ Applying this to the derivatives above, and using the fact that $(\Phi_* Y)_{\Phi(p)} = d\Phi_p Y_p = Y^j(p)\frac{\partial \Phi^i}{\partial x^j}(p)\frac{\partial}{\partial x^i}\big|_{\Phi(p)}$, with some abuse of notation, we get: \begin{align*} \nabla_{X}\left(\Phi_* Y\right) &= \left(X^j \frac{\partial}{\partial x^j} \left(Y^k \frac{\partial \Phi^i}{\partial x^k}\right)\right)\frac{\partial}{\partial x^i} \\ &= X^j \frac{\partial Y^k}{\partial x^j} \frac{\partial \Phi^i}{\partial x^k} \frac{\partial}{\partial x^i} + X^j Y^k \frac{\partial^2 \Phi^i}{\partial x^j \partial x^k} \frac{\partial}{\partial x^i} \\ &= \Phi_* \left(\nabla_X Y\right) + (\nabla_X(d\Phi))Y \end{align*} where formally $\nabla_X(d\Phi) : T_p\mathbb R^n \to T_{\Phi(p)}\mathbb R^n$ is the operator given in coordinates by $$ \nabla_X(d\Phi) = \left(X^j \frac{\partial^2 \Phi^i}{\partial x^j \partial x^k}\right)_{1 \leq i,k\leq n} $$ which by inspection roughly corresponds to a coordinatewise derivative of the Jacobian matrix of $d\Phi$ in the direction of the vector field $X$. At a point $p \in \mathbb R^n$, we can express the equation $\nabla_X \left(\Phi_* Y\right) = \Phi_*(\nabla_X Y) + (\nabla_X(d\Phi))Y$ by writing $$ \nabla_{X_{\Phi(p)}}\left(d\Phi_p Y_p\right) = d\Phi_p\left(\nabla_X Y\right)_p + (\nabla_X(d\Phi))_p Y_p $$ which seems very similar to the Leibniz "product rule".
So this all works in $\mathbb R^n$, but what about more general Riemannian manifolds? Well, formally, given $X, Y$ smooth vector fields and $\Phi : M \to M$ a diffeomorphism, we could simply define $\nabla_X(d\Phi)(Y) = \nabla_X(\Phi_*Y) - \Phi_*(\nabla_X Y)$, but this seems quite artificial. Is there a more natural/canonical way to define the covariant derivative of a diffeomorphism or its differential?